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Singularity formation for Burgers equation with transverse viscosity
TLDR
In this paper, the authors consider Burgers equation with transverse viscosity and construct a family of solutions which become singular in finite time by having their gradient becoming unbounded.Abstract:
We consider Burgers equation with transverse viscosity $$\partial_tu+u\partial_xu-\partial_{yy}u=0, \ \ (x,y)\in \mathbb R^2, \ \ u:[0,T)\times \mathbb R^2\rightarrow \mathbb R.$$ We construct and describe precisely a family of solutions which become singular in finite time by having their gradient becoming unbounded. To leading order, the solution is given by a backward self-similar solution of Burgers equation along the $x$ variable, whose scaling parameters evolve according to parabolic equations along the $y$ variable, one of them being the quadratic semi-linear heat equation. We develop a new framework adapted to this mixed hyperbolic/parabolic blow-up problem, revisit the construction of flat blow-up profiles for the semi-linear heat equation, and the self-similarity in the shocks of Burgers equation.read more
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Formation of shocks for 2D isentropic compressible Euler
TL;DR: In this article, the authors considered the 2D isentropic compressible Euler equations with pressure law and provided an elementary constructive proof of shock formation from smooth initial datum of finite energy, with no vacuum regions, and with nontrivial vorticity.
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On the implosion of a three dimensional compressible fluid
TL;DR: In this paper, the authors considered the compressible three dimensional Navier Stokes and Euler equations and constructed a set of finite energy smooth initial data for which the corresponding solutions to both equations implode (with infinite density) at a later time at a point, and completely describe the associated formation of singularity.
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Formation of point shocks for 3D compressible Euler
TL;DR: In this paper, the authors considered the 3D isentropic compressible Euler equations with the ideal gas law, and provided a constructive proof of shock formation from smooth initial data, with no vacuum regions, with nontrivial vorticity present at the shock, and under no symmetry assumptions.
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Finite time blowup of 2D Boussinesq and 3D Euler equations with $C^{1,\alpha}$ velocity and boundary
Jiajie Chen,Thomas Y. Hou +1 more
TL;DR: In this paper, Wang et al. proved the finite time singularity for the 2D Boussinesq and the 3D axisymmetric Euler equations in the presence of boundary with $C^{1,\alpha}$ initial data for the velocity field.
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Finite time blowup of 2D Boussinesq and 3D Euler equations with $C^{1,\alpha}$ velocity and boundary
Jiajie Chen,Thomas Y. Hou +1 more
TL;DR: In this paper, the authors proved the finite time singularity for the 2D Boussinesq and the 3D axisymmetric Euler equations in the presence of boundary with initial data for the velocity and density.
References
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Book
Hyberbolic Conservation Laws in Continuum Physics
TL;DR: In this paper, the authors present a masterly exposition and an encyclopedic presentation of the theory of hyperbolic conservation laws, with a focus on balance laws with dissipative source, modeling relaxation phenomena.
Book
Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States
Pavol Quittner,Philippe Souplet +1 more
TL;DR: In this article, the authors propose a model for solving the model elliptic problems and model parabolic problems. But their model is based on Equations with Gradient Terms (EGS).
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Asymptotically self‐similar blow‐up of semilinear heat equations
Yoshikazu Giga,Robert V. Kohn +1 more
TL;DR: In this paper, the authors studied the blow-up of solutions of a nonlinear heat equation and characterized the asymptotic behavior of u near a singularity, assuming a suitable upper bound on the rate of blowup.
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